Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set ${\displaystyle A}$ together with a reflexive and transitive binary relation ${\displaystyle \,\leq \,}$ (that is, a preorder), with the additional property that every pair of elements has an upper bound.^[1] In other words, for any ${\displaystyle a}$ and ${\displaystyle b}$ in ${\displaystyle A}$ there must exist ${\displaystyle c}$ in ${\displaystyle A}$ with ${\displaystyle a\leq c}$ and ${\displaystyle b\leq c.}$ A directed set's preorder is called a direction.

The notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously,^[2] meaning that every pair of elements is bounded below.^[3] Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward.^[4]

Directed sets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward.

In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets also give rise to direct limits in abstract algebra and (more generally) category theory.